Rep. Devin Nunes, California Republican and chairman of the House Permanent Select Committee on Intelligence, said Monday that he has not seen a scrap of evidence that Trump campaign officials had any contact with Russian officials, but his assurances failed to silence the growing drumbeat of calls for a special prosecutor.

Rep. Darrell E. Issa of California, a former chairman of the House Oversight and Government Reform Committee, became the most prominent Republican to join Democrats in demanding the appointment of a special prosecutor to investigate the former Cold War foe’s meddling in the presidential election and an alleged Trump campaign connection to Russian intelligence agents.

At every turn now, the president and White House officials are peppered with questions about whether anyone on the campaign had conversations with Russians and whether a special prosecutor was justified.

President Trump balked at the suggestion of an outside investigation, and Capitol Hill Republican leaders pushed back against the idea, which would threaten to hamstring the administration with an open-end probe capable of metastasizing as did Kenneth W. Starr’s investigation of President Clinton.

“I haven’t called Russia in 10 years,” Mr. Trump told executives from health insurance companies after reporters asked about the need for a special prosecutor.

The president didn’t directly respond to the reporters, who were being escorted out of the meeting at the White House, but mouthed the word “no” to the assembled guests.

Mr. Trump has asserted that he had no contact with Russians, owes no debt to Russian banks and has no business interests in Russia — all accusations made at one time or another against the president.

House Speaker Paul D. Ryan, Wisconsin Republican, said the investigations underway in the House and Senate intelligence committees were the proper venues to answer questions about Russian interference in the election.

Mr. Ryan stressed that the director of national intelligence under President Obama began the investigation during the campaign last year, when Russia was fingered for hacking into email at the Democratic National Committee and Hillary Clinton’s campaign in an attempt to hurt her White House bid.

The allegations dogged Mr. Trump throughout the campaign, including questions about his praise of Russian President Vladimir Putin, whether he had financial interests in Russia and about a discredited Russian dossier containing compromising information about him.

“We know that Russia meddled in the election. No one is disputing that,” Mr. Ryan told reporters at the White House after a meeting with Mr. Trump. “No one is alleging that some American was in on it beforehand.”

However, that is exactly what news reports continue to imply. The allegation first appeared this month in a New York Times article, which cited anonymous sources saying at least three Trump campaign officials had repeated contacts with Russian spies in the year before the election.

The report came on the heels of Michael Flynn’s forced resignation as Mr. Trump’s national security adviser after revelations that he mislead Vice President Mike Pence about conversations during the transition with the Russian ambassador about U.S. sanctions against Moscow.

The White House insisted that Mr. Flynn’s talks with the Russian ambassador to the U.S. was part of his duties but that his failure to accurately relate the content of those calls to Mr. Pence resulted in a lack of trust that cost him his job.

Mr. Nunes said he is interested in evidence of contacts between the Trump campaign and Russian spies but it doesn’t appear the investigative agencies have it.

“We still don’t have any evidence of them talking to Russians,” he told reporters at the Capitol. “As of right now, the initial inquiries I’ve made to the appropriate agencies, I don’t have any evidence.”

He said his committee has had a “long, ongoing” probe of Russian activities, which he had hoped wouldn’t become public. But based on news reports, he was forced to make the probe public and to expand it to include accusations against the Trump team.

“The way it sounds like to me is it’s been looked into and there’s no evidence of anything there,” he said.

Rep. Adam B. Schiff of California, the top Democrat on the committee, countered Mr. Nunes’s claims, saying the panel hasn’t gone deep enough to make any judgments at this point.

“The committee has reached no conclusion on whether or not the Trump campaign colluded,” Mr. Schiff said.

He didn’t contradict the facts presented by Mr. Nunes but said the committee hasn’t received any documents or interview transcripts, and has not interviewed any witnesses.

Mr. Issa also gave bipartisan credence to Democrats’ push for a special prosecutor.

He argued that Attorney General Jeff Sessions, an early supporter of Mr. Trump’s campaign, was too close to the president to oversee the FBI investigation.

“You’re going to need to use the special prosecutor’s statute and office,” Mr. Issa told host Bill Maher on Friday during HBO’s “Real Time.”

Mr. Sessions said Monday that while he had not examined in detail recent interactions between top FBI officials and the White House, the fact that contact occurred between law enforcement agencies and the White House was not inherently improper.

“The FBI and Department of Justice have to remain independent, and they will do so,” Mr. Sessions said Monday during a brief question and answer session with reporters at the Justice Department. “But every contact is not improper.”

He also said he hasn’t decided whether to recuse himself from any potential Justice Department probe.

“I would recuse myself on anything I should recuse myself on; that’s all I can tell you,” Mr. Sessions said.

“I don’t think the ghost of Ken Starr would be invoked in what is a bipartisan call to look into foreign interference in the election,” he said. “A foreign power interfered in what is the fundamental freedom of the United States. The American people deserve to know the answer here.”

Mr. Nunes‘ claims also were tainted by reports that White House press secretary Sean Spicer asked him to knock down the allegations.

Mr. Spicer said FBI officials first informed the White House that the New York Times story wasn’t accurate and the administration acted appropriately in attempting to get that information to other reporters.

“Russia’s involvement and activity has been investigated up and down. So the question becomes at some point, if there’s nothing to further investigate, what are you asking people to investigate,” he said at the daily White House press briefing.

However, when pressed by reporters about whether he would categorically deny that anyone on the Trump campaign had contact with Russians, Mr. Spicer said he couldn’t deny it.

He continued: “I guess my point is, is that you’ve had the intelligence community look at Russia’s involvement in the election. You had the House and the Senate both do the same,” he said. “And so, what I’m trying to ascertain is at what point — how many people have to say that there’s nothing there before you realize there’s nothing there?”